L\'{e}vy-walk-like Langevin dynamics affected by a time-dependent force

TL;DR

This paper models Levy walk superdiffusion under time-dependent periodic forces using a Langevin system with a subordinator, analyzing external force effects on particle dynamics and deriving a generalized Klein-Kramers equation.

Contribution

It introduces a Langevin system coupled with a subordinator to describe Levy walks in time-dependent force fields, providing new insights into external force effects on superdiffusive behavior.

Findings

External forces induce a nonzero first moment in particle displacement.

External forces add dispersion to particle positions.

The ensemble- and time-averaged mean-squared displacement are affected by external forces.

Abstract

L\'{e}vy walk is a popular and more `physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influences of external potentials almost at anytime and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the L\'{e}vy walk in the time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, the consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for space-dependent one.